Digital Discovery of 100 diverse Quantum Experiments with PyTheus

Carlos Ruiz-Gonzalez1, Sören Arlt1, Jan Petermann1, Sharareh Sayyad1, Tareq Jaouni2, Ebrahim Karimi1,2, Nora Tischler3, Xuemei Gu1, and Mario Krenn1

1Max Planck Institute for the Science of Light, Erlangen, Germany.
2Nexus for Quantum Technologies, University of Ottawa, K1N 6N5, ON, Ottawa, Canada.
3Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dynamics, Griffith University, Brisbane, Australia.

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Photons are the physical system of choice for performing experimental tests of the foundations of quantum mechanics. Furthermore, photonic quantum technology is a main player in the second quantum revolution, promising the development of better sensors, secure communications, and quantum-enhanced computation. These endeavors require generating specific quantum states or efficiently performing quantum tasks. The design of the corresponding optical experiments was historically powered by human creativity but is recently being automated with advanced computer algorithms and artificial intelligence. While several computer-designed experiments have been experimentally realized, this approach has not yet been widely adopted by the broader photonic quantum optics community. The main roadblocks consist of most systems being closed-source, inefficient, or targeted to very specific use-cases that are difficult to generalize. Here, we overcome these problems with a highly-efficient, open-source digital discovery framework PyTheus, which can employ a wide range of experimental devices from modern quantum labs to solve various tasks. This includes the discovery of highly entangled quantum states, quantum measurement schemes, quantum communication protocols, multi-particle quantum gates, as well as the optimization of continuous and discrete properties of quantum experiments or quantum states. PyTheus produces interpretable designs for complex experimental problems which human researchers can often readily conceptualize. PyTheus is an example of a powerful framework that can lead to scientific discoveries – one of the core goals of artificial intelligence in science. We hope it will help accelerate the development of quantum optics and provide new ideas in quantum hardware and technology.

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[1] Jian-Wei Pan, Zeng-Bing Chen, Chao-Yang Lu, Harald Weinfurter, Anton Zeilinger, and Marek Żukowski. Multiphoton entanglement and interferometry. Rev. Mod. Phys., 84, May 2012. 10.1103/​RevModPhys.84.777.

[2] Sheng-Kai Liao, Wen-Qi Cai, Wei-Yue Liu, Liang Zhang, Yang Li, Ji-Gang Ren, Juan Yin, Qi Shen, Yuan Cao, Zheng-Ping Li, et al. Satellite-to-ground quantum key distribution. Nature, 549 (7670), 2017. 10.1038/​nature23655.

[3] Sheng-Kai Liao, Wen-Qi Cai, Johannes Handsteiner, Bo Liu, Juan Yin, Liang Zhang, Dominik Rauch, Matthias Fink, Ji-Gang Ren, Wei-Yue Liu, et al. Satellite-relayed intercontinental quantum network. Phys. Rev. Lett., 120, Jan 2018. 10.1103/​PhysRevLett.120.030501.

[4] Bas Hensen, Hannes Bernien, Anaïs E Dréau, Andreas Reiserer, Norbert Kalb, Machiel S Blok, Just Ruitenberg, Raymond FL Vermeulen, Raymond N Schouten, Carlos Abellán, et al. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature, 526 (7575), 2015. 10.1038/​nature15759.

[5] Lynden K Shalm, Evan Meyer-Scott, Bradley G Christensen, Peter Bierhorst, Michael A Wayne, Martin J Stevens, Thomas Gerrits, Scott Glancy, Deny R Hamel, Michael S Allman, et al. Strong loophole-free test of local realism. Phys. Rev. Lett., 115, Dec 2015. 10.1103/​PhysRevLett.115.250402.

[6] Marissa Giustina, Marijn AM Versteegh, Sören Wengerowsky, Johannes Handsteiner, Armin Hochrainer, Kevin Phelan, Fabian Steinlechner, Johannes Kofler, Jan-Åke Larsson, Carlos Abellán, et al. Significant-loophole-free test of Bell's theorem with entangled photons. Phys. Rev. Lett., 115, Dec 2015. 10.1103/​PhysRevLett.115.250401.

[7] Sara Bartolucci, Patrick Birchall, Hector Bombin, Hugo Cable, Chris Dawson, Mercedes Gimeno-Segovia, Eric Johnston, Konrad Kieling, Naomi Nickerson, Mihir Pant, et al. Fusion-based quantum computation. arXiv, 2021. 10.48550/​arXiv.2101.09310.

[8] Emanuele Polino, Mauro Valeri, Nicolò Spagnolo, and Fabio Sciarrino. Photonic quantum metrology. AVS Quantum Science, 2 (2), 2020. 10.1116/​5.0007577.

[9] Christoph Schaeff, Robert Polster, Marcus Huber, Sven Ramelow, and Anton Zeilinger. Experimental access to higher-dimensional entangled quantum systems using integrated optics. Optica, 2 (6), 2015. 10.1364/​OPTICA.2.000523.

[10] Jianwei Wang, Stefano Paesani, Yunhong Ding, Raffaele Santagati, Paul Skrzypczyk, Alexia Salavrakos, Jordi Tura, Remigiusz Augusiak, Laura Mančinska, Davide Bacco, et al. Multidimensional quantum entanglement with large-scale integrated optics. Science, 360 (6386), 2018a. 10.1126/​science.aar7053.

[11] Jianwei Wang, Fabio Sciarrino, Anthony Laing, and Mark G Thompson. Integrated photonic quantum technologies. Nature Photonics, 14 (5), 2020. 10.1038/​s41566-019-0532-1.

[12] Emanuele Pelucchi, Giorgos Fagas, Igor Aharonovich, Dirk Englund, Eden Figueroa, Qihuang Gong, Hübel Hannes, Jin Liu, Chao-Yang Lu, Nobuyuki Matsuda, et al. The potential and global outlook of integrated photonics for quantum technologies. Nature Reviews Physics, 4 (3), 2022. 10.1038/​s42254-021-00398-z.

[13] Hui Wang, Yu-Ming He, T-H Chung, Hai Hu, Ying Yu, Si Chen, Xing Ding, M-C Chen, Jian Qin, Xiaoxia Yang, et al. Towards optimal single-photon sources from polarized microcavities. Nature Photonics, 13 (11), 2019. 10.1038/​s41566-019-0494-3.

[14] Yasuhiko Arakawa and Mark J Holmes. Progress in quantum-dot single photon sources for quantum information technologies: A broad spectrum overview. Applied Physics Reviews, 7 (2), 2020. 10.1063/​5.0010193.

[15] Natasha Tomm, Alisa Javadi, Nadia Olympia Antoniadis, Daniel Najer, Matthias Christian Löbl, Alexander Rolf Korsch, Rüdiger Schott, Sascha René Valentin, Andreas Dirk Wieck, Arne Ludwig, et al. A bright and fast source of coherent single photons. Nature Nanotechnology, 16 (4), 2021. 10.1038/​s41565-020-00831-x.

[16] Ravitej Uppu, Leonardo Midolo, Xiaoyan Zhou, Jacques Carolan, and Peter Lodahl. Quantum-dot-based deterministic photon–emitter interfaces for scalable photonic quantum technology. Nature nanotechnology, 16 (12), 2021. 10.1038/​s41565-021-00965-6.

[17] Tomás Santiago-Cruz, Sylvain D Gennaro, Oleg Mitrofanov, Sadhvikas Addamane, John Reno, Igal Brener, and Maria V Chekhova. Resonant metasurfaces for generating complex quantum states. Science, 377 (6609), 2022. 10.1126/​science.abq8684.

[18] Matthew D Eisaman, Jingyun Fan, Alan Migdall, and Sergey V Polyakov. Invited review article: Single-photon sources and detectors. Review of scientific instruments, 82 (7), 2011. 10.1063/​1.3610677.

[19] Sergei Slussarenko and Geoff J Pryde. Photonic quantum information processing: A concise review. Applied Physics Reviews, 6 (4), 2019. 10.1063/​1.5115814.

[20] Frédéric Bouchard, Alicia Sit, Yingwen Zhang, Robert Fickler, Filippo M Miatto, Yuan Yao, Fabio Sciarrino, and Ebrahim Karimi. Two-photon interference: the hong–ou–mandel effect. Reports on Progress in Physics, 84 (1), 2020. 10.1088/​1361-6633/​abcd7a.

[21] Adrian J. Menssen, Alex E. Jones, Benjamin J. Metcalf, Malte C. Tichy, Stefanie Barz, W. Steven Kolthammer, and Ian A. Walmsley. Distinguishability and many-particle interference. Phys. Rev. Lett., 118, Apr 2017. 10.1103/​PhysRevLett.118.153603.

[22] Lan-Tian Feng, Ming Zhang, Di Liu, Yu-Jie Cheng, Guo-Ping Guo, Dao-Xin Dai, Guang-Can Guo, Mario Krenn, and Xi-Feng Ren. On-chip quantum interference between the origins of a multi-photon state. Optica, 10 (1), 2023. 10.1364/​OPTICA.474750.

[23] Kaiyi Qian, Kai Wang, Leizhen Chen, Zhaohua Hou, Mario Krenn, Shining Zhu, and Xiao-song Ma. Multiphoton non-local quantum interference controlled by an undetected photon. Nature Communications, 14 (1), 2023. 10.1038/​s41467-023-37228-y.

[24] Mario Krenn, Manuel Erhard, and Anton Zeilinger. Computer-inspired quantum experiments. Nature Reviews Physics, 2 (11), 2020. 10.1038/​s42254-020-0230-4.

[25] Mario Krenn, Mehul Malik, Robert Fickler, Radek Lapkiewicz, and Anton Zeilinger. Automated search for new quantum experiments. Phys. Rev. Lett., 116, Mar 2016. 10.1103/​PhysRevLett.116.090405.

[26] Amin Babazadeh, Manuel Erhard, Feiran Wang, Mehul Malik, Rahman Nouroozi, Mario Krenn, and Anton Zeilinger. High-dimensional single-photon quantum gates: Concepts and experiments. Phys. Rev. Lett., 119, Nov 2017. 10.1103/​PhysRevLett.119.180510.

[27] Mehul Malik, Manuel Erhard, Marcus Huber, Mario Krenn, Robert Fickler, and Anton Zeilinger. Multi-photon entanglement in high dimensions. Nature Photonics, 10, 2016. 10.1038/​nphoton.2016.12.

[28] Manuel Erhard, Mehul Malik, Mario Krenn, and Anton Zeilinger. Experimental Greenberger–Horne–Zeilinger entanglement beyond qubits. Nature Photonics, 12 (12), 2018. 10.1038/​s41566-018-0257-6.

[29] Jaroslav Kysela, Manuel Erhard, Armin Hochrainer, Mario Krenn, and Anton Zeilinger. Path identity as a source of high-dimensional entanglement. Proceedings of the National Academy of Sciences, 117 (42), 2020. 10.1073/​pnas.2011405117.

[30] Mario Krenn, Armin Hochrainer, Mayukh Lahiri, and Anton Zeilinger. Entanglement by path identity. Phys. Rev. Lett., 118, Feb 2017a. 10.1103/​PhysRevLett.118.080401.

[31] Xiaoqin Gao, Manuel Erhard, Anton Zeilinger, and Mario Krenn. Computer-inspired concept for high-dimensional multipartite quantum gates. Phys. Rev. Lett., 125, Jul 2020. 10.1103/​PhysRevLett.125.050501.

[32] Mario Krenn, Jakob S. Kottmann, Nora Tischler, and Alán Aspuru-Guzik. Conceptual understanding through efficient automated design of quantum optical experiments. Phys. Rev. X, 11, Aug 2021. 10.1103/​PhysRevX.11.031044.

[33] Mario Krenn, Xuemei Gu, and Anton Zeilinger. Quantum experiments and graphs: Multiparty states as coherent superpositions of perfect matchings. Phys. Rev. Lett., 119, Dec 2017b. 10.1103/​PhysRevLett.119.240403.

[34] Xuemei Gu, Manuel Erhard, Anton Zeilinger, and Mario Krenn. Quantum experiments and graphs ii: Quantum interference, computation, and state generation. Proceedings of the National Academy of Sciences, 116, 2019a. 10.1073/​pnas.1815884116.

[35] Xuemei Gu, Lijun Chen, Anton Zeilinger, and Mario Krenn. Quantum experiments and graphs. iii. high-dimensional and multiparticle entanglement. Phys. Rev. A, 99, Mar 2019b. 10.1103/​PhysRevA.99.032338.

[36] Robert Raussendorf and Hans J. Briegel. A one-way quantum computer. Phys. Rev. Lett., 86, May 2001. 10.1103/​PhysRevLett.86.5188.

[37] Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel. Measurement-based quantum computation on cluster states. Phys. Rev. A, 68, Aug 2003. 10.1103/​PhysRevA.68.022312.

[38] Hans J Briegel, David E Browne, Wolfgang Dür, Robert Raussendorf, and Maarten Van den Nest. Measurement-based quantum computation. Nature Physics, 5 (1), 2009. 10.1038/​nphys1157.

[39] Sören Arlt, Carlos Ruiz-Gonzalez, and Mario Krenn. Digital discovery of a scientific concept at the core of experimental quantum optics. arXiv, 2022. 10.48550/​arXiv.2210.09981.

[40] Mario Krenn, Jonas Landgraf, Thomas Foesel, and Florian Marquardt. Artificial intelligence and machine learning for quantum technologies. Physical Review A, 107 (1), 2023. 10.1103/​PhysRevA.107.010101.

[41] PA Knott. A search algorithm for quantum state engineering and metrology. New Journal of Physics, 18 (7), 2016. 10.1088/​1367-2630/​18/​7/​073033.

[42] L O’Driscoll, Rosanna Nichols, and Paul A Knott. A hybrid machine learning algorithm for designing quantum experiments. Quantum Machine Intelligence, 1 (1), 2019. 10.1007/​s42484-019-00003-8.

[43] Rosanna Nichols, Lana Mineh, Jesús Rubio, Jonathan CF Matthews, and Paul A Knott. Designing quantum experiments with a genetic algorithm. Quantum Science and Technology, 4 (4), 2019. 10.1088/​2058-9565/​ab4d89.

[44] Xiang Zhan, Kunkun Wang, Lei Xiao, Zhihao Bian, Yongsheng Zhang, Barry C Sanders, Chengjie Zhang, and Peng Xue. Experimental quantum cloning in a pseudo-unitary system. Physical Review A, 101 (1), 2020. 10.1103/​PhysRevA.101.010302.

[45] Alexey A Melnikov, Hendrik Poulsen Nautrup, Mario Krenn, Vedran Dunjko, Markus Tiersch, Anton Zeilinger, and Hans J Briegel. Active learning machine learns to create new quantum experiments. Proceedings of the National Academy of Sciences, 115 (6), 2018. 10.1073/​pnas.1714936115.

[46] Alexey A. Melnikov, Pavel Sekatski, and Nicolas Sangouard. Setting up experimental Bell tests with reinforcement learning. Phys. Rev. Lett., 125, Oct 2020. 10.1103/​PhysRevLett.125.160401.

[47] Julius Wallnöfer, Alexey A. Melnikov, Wolfgang Dür, and Hans J. Briegel. Machine learning for long-distance quantum communication. PRX Quantum, 1, Sep 2020. 10.1103/​PRXQuantum.1.010301.

[48] X. Valcarce, P. Sekatski, E. Gouzien, A. Melnikov, and N. Sangouard. Automated design of quantum-optical experiments for device-independent quantum key distribution. Phys. Rev. A, 107, Jun 2023. 10.1103/​PhysRevA.107.062607.

[49] Thomas Adler, Manuel Erhard, Mario Krenn, Johannes Brandstetter, Johannes Kofler, and Sepp Hochreiter. Quantum optical experiments modeled by long short-term memory. In Photonics, volume 8. Multidisciplinary Digital Publishing Institute, 2021. 10.3390/​photonics8120535.

[50] Daniel Flam-Shepherd, Tony C Wu, Xuemei Gu, Alba Cervera-Lierta, Mario Krenn, and Alan Aspuru-Guzik. Learning interpretable representations of entanglement in quantum optics experiments using deep generative models. Nature Machine Intelligence, 4 (6), 2022. 10.1038/​s42256-022-00493-5.

[51] Alba Cervera-Lierta, Mario Krenn, and Alán Aspuru-Guzik. Design of quantum optical experiments with logic artificial intelligence. Quantum, 6, 2022a. 10.22331/​q-2022-10-13-836.

[52] Juan Miguel Arrazola, Thomas R Bromley, Josh Izaac, Casey R Myers, Kamil Brádler, and Nathan Killoran. Machine learning method for state preparation and gate synthesis on photonic quantum computers. Quantum Science and Technology, 4 (2), 2019. 10.1088/​2058-9565/​aaf59e.

[53] Nathan Killoran, Josh Izaac, Nicolás Quesada, Ville Bergholm, Matthew Amy, and Christian Weedbrook. Strawberry Fields: A Software Platform for Photonic Quantum Computing. Quantum, 3, Mar 2019. ISSN 2521-327X. 10.22331/​q-2019-03-11-129.

[54] Nadia Belabas, Boris Bourdoncle, Pierre-Emmanuel Emeriau, Andreas Fyrillas, Grégoire de Gliniasty, Nicolas Heurtel, Raphaël Le Bihan, Sébastien Malherbe, Rawad Mezher, Shane Mansfield, Luka Music, Marceau Paillhas, Jean Senellart, Pascale Senellart, Mario Valdiva, and Benoît Valiron. Perceval: an open source framework for programming photonic quantum computers, 2022. URL https:/​/​​Quandela/​Perceval.

[55] Budapest Quantum Computing Group. Piquasso: a python library for designing and simulating photonic quantum computers, 2022. URL https:/​/​​Budapest-Quantum-Computing-Group/​piquasso.

[56] Brajesh Gupt, Josh Izaac, and Nicolás Quesada. The walrus: a library for the calculation of hafnians, hermite polynomials and gaussian boson sampling. Journal of Open Source Software, 4 (44), 2019. 10.21105/​joss.01705.

[57] Jakob S Kottmann, Mario Krenn, Thi Ha Kyaw, Sumner Alperin-Lea, and Alán Aspuru-Guzik. Quantum computer-aided design of quantum optics hardware. Quantum Science and Technology, 6 (3), 2021. 10.1088/​2058-9565/​abfc94.

[58] Jueming Bao, Zhaorong Fu, Tanumoy Pramanik, Jun Mao, Yulin Chi, Yingkang Cao, Chonghao Zhai, Yifei Mao, Tianxiang Dai, Xiaojiong Chen, et al. Very-large-scale integrated quantum graph photonics. Nature Photonics, 17, 2023. 10.1038/​s41566-023-01187-z.

[59] Paul G. Kwiat, Klaus Mattle, Harald Weinfurter, Anton Zeilinger, Alexander V. Sergienko, and Yanhua Shih. New high-intensity source of polarization-entangled photon pairs. Phys. Rev. Lett., 75, Dec 1995. 10.1103/​PhysRevLett.75.4337.

[60] Liangliang Lu, Lijun Xia, Zhiyu Chen, Leizhen Chen, Tonghua Yu, Tao Tao, Wenchao Ma, Ying Pan, Xinlun Cai, Yanqing Lu, et al. Three-dimensional entanglement on a silicon chip. npj Quantum Information, 6 (1), 2020. 10.1038/​s41534-020-0260-x.

[61] Halina Rubinsztein-Dunlop, Andrew Forbes, Michael V Berry, Mark R Dennis, David L Andrews, Masud Mansuripur, Cornelia Denz, Christina Alpmann, Peter Banzer, Thomas Bauer, et al. Roadmap on structured light. Journal of Optics, 19 (1), 2016. 10.1088/​2040-8978/​19/​1/​013001.

[62] Miles J Padgett. Orbital angular momentum 25 years on. Optics express, 25 (10), 2017. 10.1364/​OE.25.011265.

[63] Frédéric Bouchard, Robert Fickler, Robert W Boyd, and Ebrahim Karimi. High-dimensional quantum cloning and applications to quantum hacking. Science Advances, 3 (2), 2017a. 10.1126/​sciadv.1601915.

[64] Jessica Bavaresco, Natalia Herrera Valencia, Claude Klöckl, Matej Pivoluska, Paul Erker, Nicolai Friis, Mehul Malik, and Marcus Huber. Measurements in two bases are sufficient for certifying high-dimensional entanglement. Nature Physics, 14 (10), 2018. 10.1038/​s41567-018-0203-z.

[65] J. D. Franson. Bell inequality for position and time. Phys. Rev. Lett., 62, May 1989. 10.1103/​PhysRevLett.62.2205.

[66] L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. Phan Huy. Frequency-bin entangled photons. Phys. Rev. A, 82, Jul 2010. 10.1103/​PhysRevA.82.013804.

[67] Robert W Boyd. Nonlinear optics, Fourth Edition. Academic press, 2020. 10.1016/​C2015-0-05510-1.

[68] Regina Kruse, Craig S. Hamilton, Linda Sansoni, Sonja Barkhofen, Christine Silberhorn, and Igor Jex. Detailed study of gaussian boson sampling. Phys. Rev. A, 100, Sep 2019. 10.1103/​PhysRevA.100.032326.

[69] Armin Hochrainer, Mayukh Lahiri, Manuel Erhard, Mario Krenn, and Anton Zeilinger. Quantum indistinguishability by path identity and with undetected photons. Rev. Mod. Phys., 94, Jun 2022. 10.1103/​RevModPhys.94.025007.

[70] Xi-Lin Wang, Luo-Kan Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, H. Lu, Y. Hu, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, Yu-Ao Chen, Chao-Yang Lu, and Jian-Wei Pan. Experimental ten-photon entanglement. Phys. Rev. Lett., 117, Nov 2016. 10.1103/​PhysRevLett.117.210502.

[71] Luo-Kan Chen, Zheng-Da Li, Xing-Can Yao, Miao Huang, Wei Li, He Lu, Xiao Yuan, Yan-Bao Zhang, Xiao Jiang, Cheng-Zhi Peng, et al. Observation of ten-photon entanglement using thin bib 3 o 6 crystals. Optica, 4 (1), 2017a. 10.1364/​OPTICA.4.000077.

[72] Paul G. Kwiat, Edo Waks, Andrew G. White, Ian Appelbaum, and Philippe H. Eberhard. Ultrabright source of polarization-entangled photons. Phys. Rev. A, 60, Aug 1999. 10.1103/​PhysRevA.60.R773.

[73] John Calsamiglia. Generalized measurements by linear elements. Phys. Rev. A, 65, Feb 2002. 10.1103/​PhysRevA.65.030301.

[74] Stefano Paesani, Jacob F. F. Bulmer, Alex E. Jones, Raffaele Santagati, and Anthony Laing. Scheme for universal high-dimensional quantum computation with linear optics. Phys. Rev. Lett., 126, Jun 2021. 10.1103/​PhysRevLett.126.230504.

[75] Seungbeom Chin, Yong-Su Kim, and Sangmin Lee. Graph picture of linear quantum networks and entanglement. Quantum, 5, 2021. 10.22331/​q-2021-12-23-611.

[76] AV Belinskii and DN Klyshko. Two-photon optics: diffraction, holography, and transformation of two-dimensional signals. Soviet Journal of Experimental and Theoretical Physics, 78 (3), 1994. URL http:/​/​​cgi-bin/​dn/​e_078_03_0259.pdf.

[77] M. F. Z. Arruda, W. C. Soares, S. P. Walborn, D. S. Tasca, A. Kanaan, R. Medeiros de Araújo, and P. H. Souto Ribeiro. Klyshko's advanced-wave picture in stimulated parametric down-conversion with a spatially structured pump beam. Phys. Rev. A, 98, Aug 2018. 10.1103/​PhysRevA.98.023850.

[78] Evan Meyer-Scott, Christine Silberhorn, and Alan Migdall. Single-photon sources: Approaching the ideal through multiplexing. Review of Scientific Instruments, 91 (4), 2020. 10.1063/​5.0003320.

[79] Barry C. Sanders. Quantum dynamics of the nonlinear rotator and the effects of continual spin measurement. Phys. Rev. A, 40, Sep 1989. 10.1103/​PhysRevA.40.2417.

[80] Hwang Lee, Pieter Kok, and Jonathan P Dowling. A quantum rosetta stone for interferometry. Journal of Modern Optics, 49 (14-15), 2002. 10.1080/​0950034021000011536.

[81] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Advances in quantum metrology. Nature photonics, 5 (4), 2011. 10.1038/​nphoton.2011.35.

[82] Lu Zhang and Kam Wai Clifford Chan. Scalable generation of multi-mode noon states for quantum multiple-phase estimation. Scientific reports, 8 (1), 2018. 10.1038/​s41598-018-29828-2.

[83] Seongjin Hong, Yong-Su Kim, Young-Wook Cho, Seung-Woo Lee, Hojoong Jung, Sung Moon, Sang-Wook Han, Hyang-Tag Lim, et al. Quantum enhanced multiple-phase estimation with multi-mode n00n states. Nature Communications, 12 (1), 2021. 10.1038/​s41467-021-25451-4.

[84] A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, D. N. Klyshko, and S. P. Kulik. Polarization state of a biphoton: Quantum ternary logic. Phys. Rev. A, 60, Dec 1999. 10.1103/​PhysRevA.60.R4209.

[85] A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, and S. P. Kulik. Collinear two-photon state with spectral properties of type-i and polarization properties of type-ii spontaneous parametric down-conversion: Preparation and testing. Phys. Rev. A, 64, Sep 2001. 10.1103/​PhysRevA.64.041803.

[86] Itai Afek, Oron Ambar, and Yaron Silberberg. High-noon states by mixing quantum and classical light. Science, 328 (5980), 2010. 10.1126/​science.1188172].

[87] C. K. Hong, Z. Y. Ou, and L. Mandel. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett., 59, Nov 1987. 10.1103/​PhysRevLett.59.2044.

[88] M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert. "event-ready-detectors" bell experiment via entanglement swapping. Phys. Rev. Lett., 71, Dec 1993. 10.1103/​PhysRevLett.71.4287.

[89] Jian-Wei Pan, Dik Bouwmeester, Harald Weinfurter, and Anton Zeilinger. Experimental entanglement swapping: Entangling photons that never interacted. Phys. Rev. Lett., 80, May 1998. 10.1103/​PhysRevLett.80.3891.

[90] Nicolas Sangouard, Christoph Simon, Hugues de Riedmatten, and Nicolas Gisin. Quantum repeaters based on atomic ensembles and linear optics. Rev. Mod. Phys., 83, Mar 2011. 10.1103/​RevModPhys.83.33.

[91] F. Basso Basset, M. B. Rota, C. Schimpf, D. Tedeschi, K. D. Zeuner, S. F. Covre da Silva, M. Reindl, V. Zwiller, K. D. Jöns, A. Rastelli, and R. Trotta. Entanglement swapping with photons generated on demand by a quantum dot. Phys. Rev. Lett., 123, Oct 2019. 10.1103/​PhysRevLett.123.160501.

[92] Daniel Llewellyn, Yunhong Ding, Imad I Faruque, Stefano Paesani, Davide Bacco, Raffaele Santagati, Yan-Jun Qian, Yan Li, Yun-Feng Xiao, Marcus Huber, et al. Chip-to-chip quantum teleportation and multi-photon entanglement in silicon. Nature Physics, 16 (2), 2020. 10.1038/​s41567-019-0727-x.

[93] Farid Samara, Nicolas Maring, Anthony Martin, Arslan S Raja, Tobias J Kippenberg, Hugo Zbinden, and Rob Thew. Entanglement swapping between independent and asynchronous integrated photon-pair sources. Quantum Science and Technology, 6 (4), 2021. 10.1088/​2058-9565/​abf599.

[94] Harald Weinfurter. Experimental Bell-state analysis. EPL (Europhysics Letters), 25 (8), 1994. 10.1209/​0295-5075/​25/​8/​001.

[95] Markus Michler, Klaus Mattle, Harald Weinfurter, and Anton Zeilinger. Interferometric Bell-state analysis. Phys. Rev. A, 53, Mar 1996. 10.1103/​PhysRevA.53.R1209.

[96] Michael A Nielsen and Isaac L Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press; 10th Anniversary edition (9 Dec. 2010), 2010. 10.1017/​CBO9780511976667.

[97] Emanuel Knill, Raymond Laflamme, and Gerald J Milburn. A scheme for efficient quantum computation with linear optics. nature, 409 (6816), 2001. 10.1038/​35051009.

[98] Sara Gasparoni, Jian-Wei Pan, Philip Walther, Terry Rudolph, and Anton Zeilinger. Realization of a photonic controlled-not gate sufficient for quantum computation. Phys. Rev. Lett., 93, Jul 2004. 10.1103/​PhysRevLett.93.020504.

[99] Pieter Kok, W. J. Munro, Kae Nemoto, T. C. Ralph, Jonathan P. Dowling, and G. J. Milburn. Linear optical quantum computing with photonic qubits. Rev. Mod. Phys., 79, Jan 2007. 10.1103/​RevModPhys.79.135.

[100] Yuan Li, Lingxiao Wan, Hui Zhang, Huihui Zhu, Yuzhi Shi, Lip Ket Chin, Xiaoqi Zhou, Leong Chuan Kwek, and Ai Qun Liu. Quantum fredkin and toffoli gates on a versatile programmable silicon photonic chip. npj Quantum Information, 8 (1), September 2022. 10.1038/​s41534-022-00627-y.

[101] E. Knill. Quantum gates using linear optics and postselection. Physical Review A, 66 (5), November 2002. 10.1103/​physreva.66.052306.

[102] T. C. Ralph, N. K. Langford, T. B. Bell, and A. G. White. Linear optical controlled-not gate in the coincidence basis. Phys. Rev. A, 65, Jun 2002. 10.1103/​PhysRevA.65.062324.

[103] J. L. O'Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning. Demonstration of an all-optical quantum controlled-NOT gate. Nature, 426, 2003. 10.1038/​nature02054.

[104] N. K. Langford, T. J. Weinhold, R. Prevedel, K. J. Resch, A. Gilchrist, J. L. O'Brien, G. J. Pryde, and A. G. White. Demonstration of a simple entangling optical gate and its use in Bell-state analysis. Phys. Rev. Lett., 95, Nov 2005. 10.1103/​PhysRevLett.95.210504.

[105] Farzad Ghafari, Nora Tischler, Jayne Thompson, Mile Gu, Lynden K. Shalm, Varun B. Verma, Sae Woo Nam, Raj B. Patel, Howard M. Wiseman, and Geoff J. Pryde. Dimensional quantum memory advantage in the simulation of stochastic processes. Phys. Rev. X, 9, Oct 2019. 10.1103/​PhysRevX.9.041013.

[106] Raj B Patel, Joseph Ho, Franck Ferreyrol, Timothy C Ralph, and Geoff J Pryde. A quantum fredkin gate. Science Advances, 2 (3), 2016. 10.1126/​sciadv.1501531.

[107] Shakib Daryanoosh, Sergei Slussarenko, Dominic W. Berry, Howard M. Wiseman, and Geoff J. Pryde. Experimental optical phase measurement approaching the exact Heisenberg limit. Nature Communications, 9, 2018. 10.1038/​s41467-018-06601-7.

[108] Zhi Zhao, An-Ning Zhang, Yu-Ao Chen, Han Zhang, Jiang-Feng Du, Tao Yang, and Jian-Wei Pan. Experimental demonstration of a nondestructive controlled-not quantum gate for two independent photon qubits. Phys. Rev. Lett., 94, Jan 2005. 10.1103/​PhysRevLett.94.030501.

[109] Xiao-Hui Bao, Teng-Yun Chen, Qiang Zhang, Jian Yang, Han Zhang, Tao Yang, and Jian-Wei Pan. Optical nondestructive controlled-not gate without using entangled photons. Phys. Rev. Lett., 98, Apr 2007. 10.1103/​PhysRevLett.98.170502.

[110] Wei-Bo Gao, Alexander M Goebel, Chao-Yang Lu, Han-Ning Dai, Claudia Wagenknecht, Qiang Zhang, Bo Zhao, Cheng-Zhi Peng, Zeng-Bing Chen, Yu-Ao Chen, et al. Teleportation-based realization of an optical quantum two-qubit entangling gate. Proceedings of the National Academy of Sciences, 107 (49), 2010. 10.1073/​pnas.1005720107.

[111] Ryo Okamoto, Jeremy L O’Brien, Holger F Hofmann, and Shigeki Takeuchi. Realization of a knill-laflamme-milburn controlled-not photonic quantum circuit combining effective optical nonlinearities. Proceedings of the National Academy of Sciences, 108 (25), 2011. 10.1073/​pnas.101883910.

[112] Jin-Peng Li, Xuemei Gu, Jian Qin, Dian Wu, Xiang You, Hui Wang, Christian Schneider, Sven Höfling, Yong-Heng Huo, Chao-Yang Lu, Nai-Le Liu, Li Li, and Jian-Wei Pan. Heralded nondestructive quantum entangling gate with single-photon sources. Phys. Rev. Lett., 126, Apr 2021. 10.1103/​PhysRevLett.126.140501.

[113] Jonas Zeuner, Aditya N. Sharma, Max Tillmann, René Heilmann, Markus Gräfe, Amir Moqanaki, Alexander Szameit, and Philip Walther. Integrated-optics heralded controlled-NOT gate for polarization-encoded qubits. npj Quantum Information, 4, 2018. 10.1038/​s41534-018-0068-0.

[114] Reuben S Aspden, Daniel S Tasca, Andrew Forbes, Robert W Boyd, and Miles J Padgett. Experimental demonstration of klyshko’s advanced-wave picture using a coincidence-count based, camera-enabled imaging system. Journal of Modern Optics, 61 (7), 2014. 10.1080/​09500340.2014.899645.

[115] Min Jiang, Shunlong Luo, and Shuangshuang Fu. Channel-state duality. Phys. Rev. A, 87, Feb 2013. 10.1103/​PhysRevA.87.022310.

[116] Jay Lawrence. Rotational covariance and Greenberger-Horne-Zeilinger theorems for three or more particles of any dimension. Phys. Rev. A, 89, Jan 2014. 10.1103/​PhysRevA.89.012105.

[117] Lev Vaidman, Yakir Aharonov, and David Z. Albert. How to ascertain the values of ${\mathrm{sigma}}_{\mathrm{x}}$, ${\mathrm{{\sigma}}}_{\mathrm{y}}$, and ${\mathrm{{\sigma}}}_{\mathrm{z}}$ of a spin-1/​2 particle. Phys. Rev. Lett., 58, Apr 1987. 10.1103/​PhysRevLett.58.1385.

[118] Asher Peres. All the Bell inequalities. Foundations of Physics, 29 (4), 1999. 10.1023/​A:1018816310000.

[119] Tobias Moroder, Oleg Gittsovich, Marcus Huber, and Otfried Gühne. Steering bound entangled states: A counterexample to the stronger peres conjecture. Phys. Rev. Lett., 113, Aug 2014. 10.1103/​PhysRevLett.113.050404.

[120] Tamás Vértesi and Nicolas Brunner. Disproving the peres conjecture by showing Bell nonlocality from bound entanglement. Nature Communications, 5 (1), 2014. 10.1038/​ncomms6297.

[121] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev., 47, May 1935. 10.1103/​PhysRev.47.777.

[122] J. S. Bell. On the einstein podolsky rosen paradox. Physics, 1, Nov 1964. 10.1103/​PhysicsPhysiqueFizika.1.195.

[123] Daniel M Greenberger, Michael A Horne, and Anton Zeilinger. Going beyond Bell’s theorem. In Bell’s theorem, quantum theory and conceptions of the universe. Springer, 1989. 10.1007/​978-94-017-0849-4_10.

[124] Daniel M Greenberger, Michael A Horne, Abner Shimony, and Anton Zeilinger. Bell’s theorem without inequalities. American Journal of Physics, 58 (12), 1990. 10.1119/​1.16243.

[125] Jian-Wei Pan, Dik Bouwmeester, Matthew Daniell, Harald Weinfurter, and Anton Zeilinger. Experimental test of quantum nonlocality in three-photon Greenberger–Horne–Zeilinger entanglement. Nature, 403 (6769), 2000. 10.1038/​35000514.

[126] Junghee Ryu, Changhyoup Lee, Zhi Yin, Ramij Rahaman, Dimitris G. Angelakis, Jinhyoung Lee, and Marek Żukowski. Multisetting Greenberger-Horne-Zeilinger theorem. Phys. Rev. A, 89, Feb 2014. 10.1103/​PhysRevA.89.024103.

[127] Jay Lawrence. Many-qutrit mermin inequalities with three measurement settings. arXiv, 2019. 10.48550/​arXiv.1910.05869.

[128] Manuel Erhard, Mario Krenn, and Anton Zeilinger. Advances in high-dimensional quantum entanglement. Nature Reviews Physics, 2 (7), 2020. 10.1038/​s42254-020-0193-5.

[129] Xi-Lin Wang, Yi-Han Luo, He-Liang Huang, Ming-Cheng Chen, Zu-En Su, Chang Liu, Chao Chen, Wei Li, Yu-Qiang Fang, Xiao Jiang, Jun Zhang, Li Li, Nai-Le Liu, Chao-Yang Lu, and Jian-Wei Pan. 18-qubit entanglement with six photons' three degrees of freedom. Phys. Rev. Lett., 120, Jun 2018b. 10.1103/​PhysRevLett.120.260502.

[130] Alba Cervera-Lierta, Mario Krenn, Alán Aspuru-Guzik, and Alexey Galda. Experimental high-dimensional greenberger-horne-zeilinger entanglement with superconducting transmon qutrits. Phys. Rev. Applied, 17, Feb 2022b. 10.1103/​PhysRevApplied.17.024062.

[131] Denis Sych and Gerd Leuchs. A complete basis of generalized Bell states. New Journal of Physics, 11 (1), 2009. 10.1088/​1367-2630/​11/​1/​013006.

[132] Gregg Jaeger. Bell gems: the Bell basis generalized. Physics Letters A, 329 (6), 2004. 10.1016/​j.physleta.2004.07.037.

[133] F. Verstraete, J. Dehaene, B. De Moor, and H. Verschelde. Four qubits can be entangled in nine different ways. Phys. Rev. A, 65, Apr 2002. 10.1103/​PhysRevA.65.052112.

[134] Peter W. Shor. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A, 52, Oct 1995. 10.1103/​PhysRevA.52.R2493.

[135] Andrew Steane. Multiple-particle interference and quantum error correction. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 452 (1954), 1996. 10.1098/​rspa.1996.0136.

[136] Raymond Laflamme, Cesar Miquel, Juan Pablo Paz, and Wojciech Hubert Zurek. Perfect quantum error correcting code. Phys. Rev. Lett., 77, Jul 1996. 10.1103/​PhysRevLett.77.198.

[137] David P. DiVincenzo and Peter W. Shor. Fault-tolerant error correction with efficient quantum codes. Phys. Rev. Lett., 77, Oct 1996. 10.1103/​PhysRevLett.77.3260.

[138] Mohamed Bourennane, Manfred Eibl, Sascha Gaertner, Nikolai Kiesel, Christian Kurtsiefer, and Harald Weinfurter. Entanglement persistency of multiphoton entangled states. Phys. Rev. Lett., 96, Mar 2006. 10.1103/​PhysRevLett.96.100502.

[139] M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral. Quantum telecloning and multiparticle entanglement. Phys. Rev. A, 59, Jan 1999. 10.1103/​PhysRevA.59.156.

[140] R. Prevedel, G. Cronenberg, M. S. Tame, M. Paternostro, P. Walther, M. S. Kim, and A. Zeilinger. Experimental realization of dicke states of up to six qubits for multiparty quantum networking. Phys. Rev. Lett., 103, Jul 2009. 10.1103/​PhysRevLett.103.020503.

[141] Luca Pezzè, Augusto Smerzi, Markus K. Oberthaler, Roman Schmied, and Philipp Treutlein. Quantum metrology with nonclassical states of atomic ensembles. Rev. Mod. Phys., 90, Sep 2018. 10.1103/​RevModPhys.90.035005.

[142] Tzu-Chieh Wei and Paul M. Goldbart. Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A, 68, Oct 2003. 10.1103/​PhysRevA.68.042307.

[143] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels. Phys. Rev. Lett., 70, 3 1993. 10.1103/​PhysRevLett.70.1895.

[144] Ye Yeo and Wee Kang Chua. Teleportation and dense coding with genuine multipartite entanglement. Phys. Rev. Lett., 96, Feb 2006. 10.1103/​PhysRevLett.96.060502.

[145] Cezary Śliwa and Konrad Banaszek. Conditional preparation of maximal polarization entanglement. Phys. Rev. A, 67, Mar 2003. 10.1103/​PhysRevA.67.030101.

[146] F. V. Gubarev, I. V. Dyakonov, M. Yu. Saygin, G. I. Struchalin, S. S. Straupe, and S. P. Kulik. Improved heralded schemes to generate entangled states from single photons. Phys. Rev. A, 102, Jul 2020. 10.1103/​PhysRevA.102.012604.

[147] Marcus Huber and Julio I. de Vicente. Structure of multidimensional entanglement in multipartite systems. Phys. Rev. Lett., 110, Jan 2013. 10.1103/​PhysRevLett.110.030501.

[148] Marcus Huber, Martí Perarnau-Llobet, and Julio I. de Vicente. Entropy vector formalism and the structure of multidimensional entanglement in multipartite systems. Phys. Rev. A, 88, Oct 2013. 10.1103/​PhysRevA.88.042328.

[149] Josh Cadney, Marcus Huber, Noah Linden, and Andreas Winter. Inequalities for the ranks of multipartite quantum states. Linear Algebra and its Applications, 452, 2014. 10.1016/​j.laa.2014.03.035.

[150] Matej Pivoluska, Marcus Huber, and Mehul Malik. Layered quantum key distribution. Phys. Rev. A, 97, Mar 2018. 10.1103/​PhysRevA.97.032312.

[151] Xuemei Gu, Lijun Chen, and Mario Krenn. Quantum experiments and hypergraphs: Multiphoton sources for quantum interference, quantum computation, and quantum entanglement. Phys. Rev. A, 101, Mar 2020. 10.1103/​PhysRevA.101.033816.

[152] Xiao-Min Hu, Wen-Bo Xing, Chao Zhang, Bi-Heng Liu, Matej Pivoluska, Marcus Huber, Yun-Feng Huang, Chuan-Feng Li, and Guang-Can Guo. Experimental creation of multi-photon high-dimensional layered quantum states. npj Quantum Information, 6 (1), 2020. 10.1038/​s41534-020-00318-6.

[153] Akimasa Miyake. Classification of multipartite entangled states by multidimensional determinants. Phys. Rev. A, 67, Jan 2003. 10.1103/​PhysRevA.67.012108.

[154] Asher Peres. Separability criterion for density matrices. Phys. Rev. Lett., 77, Aug 1996. 10.1103/​PhysRevLett.77.1413.

[155] Michał Horodecki. Entanglement measures. Quantum Information & Computation, 1 (1), 2001. 10.5555/​2011326.2011328.

[156] Iain DK Brown, Susan Stepney, Anthony Sudbery, and Samuel L Braunstein. Searching for highly entangled multi-qubit states. Journal of Physics A: Mathematical and General, 38 (5), 2005. 10.1088/​0305-4470/​38/​5/​013.

[157] Alfréd Rényi et al. On measures of entropy and information. In Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, 1961. URL http:/​/​​Horticulture/​GAs/​Refs/​Renyi_1961.pdf.

[158] Wim Van Dam and Patrick Hayden. Renyi-entropic bounds on quantum communication. arXiv, 2002. 10.48550/​arXiv.quant-ph/​0204093.

[159] Gilad Gour and Nolan R Wallach. All maximally entangled four-qubit states. Journal of Mathematical Physics, 51 (11), 2010. 10.1063/​1.3511477.

[160] Gavin K. Brennen. An observable measure of entanglement for pure states of multi-qubit systems. Quantum Inf. Comput., 3 (6), 2003. 10.26421/​QIC3.6-5.

[161] David A Meyer and Nolan R Wallach. Global entanglement in multiparticle systems. Journal of Mathematical Physics, 43 (9), 2002. 10.1063/​1.1497700.

[162] Marco Enríquez, Zbigniew Puchała, and Karol Życzkowski. Minimal rényi–ingarden–urbanik entropy of multipartite quantum states. Entropy, 17 (7), 2015. 10.3390/​e17075063.

[163] Wolfram Helwig. Absolutely maximally entangled qudit graph states. arXiv, 2013. 10.48550/​arXiv.1306.2879.

[164] Dardo Goyeneche and Karol Życzkowski. Genuinely multipartite entangled states and orthogonal arrays. Phys. Rev. A, 90, Aug 2014. 10.1103/​PhysRevA.90.022316.

[165] Fei Shi, Yi Shen, Lin Chen, and Xiande Zhang. Constructions of ${k}$-uniform states from mixed orthogonal arrays. arXiv, 2020. 10.48550/​arXiv.2006.04086.

[166] A. Higuchi and A. Sudbery. How entangled can two couples get? Physics Letters A, 273 (4), August 2000. 10.1016/​s0375-9601(00)00480-1.

[167] Lucien Hardy. Nonlocality for two particles without inequalities for almost all entangled states. Phys. Rev. Lett., 71, Sep 1993. 10.1103/​PhysRevLett.71.1665.

[168] Lixiang Chen, Wuhong Zhang, Ziwen Wu, Jikang Wang, Robert Fickler, and Ebrahim Karimi. Experimental ladder proof of hardy's nonlocality for high-dimensional quantum systems. Phys. Rev. A, 96, Aug 2017b. 10.1103/​PhysRevA.96.022115.

[169] Kishor Bharti, Tobias Haug, Vlatko Vedral, and Leong-Chuan Kwek. Machine learning meets quantum foundations: A brief survey. AVS Quantum Science, 2 (3), 2020. 10.1116/​5.0007529.

[170] Joseph Bowles, Flavien Hirsch, and Daniel Cavalcanti. Single-copy activation of Bell nonlocality via broadcasting of quantum states. Quantum, 5, jul 2021. ISSN 2521-327X. 10.22331/​q-2021-07-13-499.

[171] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum-enhanced measurements: beating the standard quantum limit. Science, 306 (5700), 2004. 10.1126/​science.1104149.

[172] Christoph F. Wildfeuer, Austin P. Lund, and Jonathan P. Dowling. Strong violations of Bell-type inequalities for path-entangled number states. Phys. Rev. A, 76, Nov 2007. 10.1103/​PhysRevA.76.052101.

[173] Yonatan Israel, Shamir Rosen, and Yaron Silberberg. Supersensitive polarization microscopy using noon states of light. Phys. Rev. Lett., 112, Mar 2014. 10.1103/​PhysRevLett.112.103604.

[174] Takafumi Ono, Ryo Okamoto, and Shigeki Takeuchi. An entanglement-enhanced microscope. Nature Communications, 4 (1), 2013. 10.1038/​ncomms3426.

[175] Xiaoqin Gao, Yingwen Zhang, Alessio D’Errico, Khabat Heshami, and Ebrahim Karimi. High-speed imaging of spatiotemporal correlations in hong-ou-mandel interference. Optics Express, 30 (11), 2022. 10.1364/​OE.456433.

[176] Bienvenu Ndagano, Hugo Defienne, Dominic Branford, Yash D Shah, Ashley Lyons, Niclas Westerberg, Erik M Gauger, and Daniele Faccio. Quantum microscopy based on hong–ou–mandel interference. Nature Photonics, 16 (5), 2022. 10.1038/​s41566-022-00980-6.

[177] Morgan W Mitchell, Jeff S Lundeen, and Aephraem M Steinberg. Super-resolving phase measurements with a multiphoton entangled state. Nature, 429 (6988), 2004. 10.1038/​nature02493.

[178] Philip Walther, Jian-Wei Pan, Markus Aspelmeyer, Rupert Ursin, Sara Gasparoni, and Anton Zeilinger. De broglie wavelength of a non-local four-photon state. Nature, 429 (6988), 2004. 10.1038/​nature02552.

[179] F. W. Sun, B. H. Liu, Y. F. Huang, Z. Y. Ou, and G. C. Guo. Observation of the four-photon de broglie wavelength by state-projection measurement. Phys. Rev. A, 74, Sep 2006. 10.1103/​PhysRevA.74.033812.

[180] K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O'Brien, and A. G. White. Time-reversal and super-resolving phase measurements. Phys. Rev. Lett., 98, May 2007. 10.1103/​PhysRevLett.98.223601.

[181] Agedi N. Boto, Pieter Kok, Daniel S. Abrams, Samuel L. Braunstein, Colin P. Williams, and Jonathan P. Dowling. Quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit. Phys. Rev. Lett., 85, Sep 2000. 10.1103/​PhysRevLett.85.2733.

[182] Erwin Schrödinger. Die gegenwärtige situation in der quantenmechanik. Naturwissenschaften, 23 (50), 1935. URL https:/​/​​solutions/​scientists/​schrodinger/​Die_Situation-3.pdf.

[183] Kishore T. Kapale and Jonathan P. Dowling. Bootstrapping approach for generating maximally path-entangled photon states. Phys. Rev. Lett., 99, Aug 2007. 10.1103/​PhysRevLett.99.053602.

[184] Hugo Cable and Jonathan P. Dowling. Efficient generation of large number-path entanglement using only linear optics and feed-forward. Phys. Rev. Lett., 99, Oct 2007. 10.1103/​PhysRevLett.99.163604.

[185] Luca Pezzé and Augusto Smerzi. Mach-zehnder interferometry at the heisenberg limit with coherent and squeezed-vacuum light. Phys. Rev. Lett., 100, Feb 2008. 10.1103/​PhysRevLett.100.073601.

[186] Holger F. Hofmann and Takafumi Ono. High-photon-number path entanglement in the interference of spontaneously down-converted photon pairs with coherent laser light. Phys. Rev. A, 76, Sep 2007. 10.1103/​PhysRevA.76.031806.

[187] Y. Israel, I. Afek, S. Rosen, O. Ambar, and Y. Silberberg. Experimental tomography of noon states with large photon numbers. Phys. Rev. A, 85, Feb 2012. 10.1103/​PhysRevA.85.022115.

[188] Peter C. Humphreys, Marco Barbieri, Animesh Datta, and Ian A. Walmsley. Quantum enhanced multiple phase estimation. Phys. Rev. Lett., 111, Aug 2013. 10.1103/​PhysRevLett.111.070403.

[189] P. A. Knott, T. J. Proctor, A. J. Hayes, J. F. Ralph, P. Kok, and J. A. Dunningham. Local versus global strategies in multiparameter estimation. Phys. Rev. A, 94, Dec 2016. 10.1103/​PhysRevA.94.062312.

[190] Heonoh Kim, Hee Su Park, and Sang-Kyung Choi. Three-photon n00n states generated by photon subtraction from double photon pairs. Optics Express, 17 (22), 2009. 10.1364/​OE.17.019720.

[191] Yosep Kim, Gunnar Björk, and Yoon-Ho Kim. Experimental characterization of quantum polarization of three-photon states. Phys. Rev. A, 96, Sep 2017. 10.1103/​PhysRevA.96.033840.

[192] Yong-Su Kim, Osung Kwon, Sang Min Lee, Jong-Chan Lee, Heonoh Kim, Sang-Kyung Choi, Hee Su Park, and Yoon-Ho Kim. Observation of young’s double-slit interference with the three-photon n00n state. Optics Express, 19 (25), 2011. 10.1364/​OE.19.024957.

[193] Gunnar Björk, Markus Grassl, Pablo de la Hoz, Gerd Leuchs, and Luis L Sánchez-Soto. Stars of the quantum universe: extremal constellations on the poincaré sphere. Physica Scripta, 90 (10), 2015. 10.1088/​0031-8949/​90/​10/​108008.

[194] G. Björk, A. B. Klimov, P. de la Hoz, M. Grassl, G. Leuchs, and L. L. Sánchez-Soto. Extremal quantum states and their majorana constellations. Phys. Rev. A, 92, Sep 2015. 10.1103/​PhysRevA.92.031801.

[195] Frederic Bouchard, P de la Hoz, Gunnar Björk, RW Boyd, Markus Grassl, Z Hradil, E Karimi, AB Klimov, Gerd Leuchs, J Řeháček, et al. Quantum metrology at the limit with extremal majorana constellations. Optica, 4 (11), 2017b. 10.1364/​OPTICA.4.001429.

[196] Ettore Majorana. Atomi orientati in campo magnetico variabile. Il Nuovo Cimento (1924-1942), 9 (2), 1932. 10.1007/​BF02960953.

[197] John H Conway, Ronald H Hardin, and Neil JA Sloane. Packing lines, planes, etc.: Packings in grassmannian spaces. Experimental mathematics, 5 (2), 1996. 10.1080/​10586458.1996.10504585.

[198] Edward B Saff and Amo BJ Kuijlaars. Distributing many points on a sphere. The mathematical intelligencer, 19 (1), 1997. 10.1007/​BF03024331.

[199] Armin Tavakoli and Nicolas Gisin. The platonic solids and fundamental tests of quantum mechanics. Quantum, 4, 2020. 10.22331/​q-2020-07-09-293.

[200] Károly F Pál and Tamás Vértesi. Platonic Bell inequalities for all dimensions. Quantum, 6, 2022. 10.22331/​q-2022-07-07-756.

[201] Markus Grassl. Extremal polarization states, 2015. URL http:/​/​​index.html.

[202] Hugo Ferretti. Quantum Parameter Estimation in the Laboratory. PhD thesis, University of Toronto (Canada), 2022. URL https:/​/​​dissertations-theses/​quantum-parameter-estimation-laboratory/​docview/​2646725686/​se-2.

[203] Alán Aspuru-Guzik and Philip Walther. Photonic quantum simulators. Nature physics, 8 (4), 2012. 10.1038/​nphys2253.

[204] Ulrich Schollwöck. The density-matrix renormalization group in the age of matrix product states. Annals of physics, 326 (1), 2011. 10.1016/​j.aop.2010.09.012.

[205] J. Ignacio Cirac, David Pérez-Garcia, Norbert Schuch, and Frank Verstraete. Matrix product states and projected entangled pair states: Concepts, symmetries, theorems. Rev. Mod. Phys., 93, Dec 2021. 10.1103/​RevModPhys.93.045003.

[206] Jorge Miguel-Ramiro and Wolfgang Dür. Delocalized information in quantum networks. New Journal of Physics, 22 (4), 2020. 10.1088/​1367-2630/​ab784d.

[207] D. Gross and J. Eisert. Quantum computational webs. Phys. Rev. A, 82, Oct 2010. 10.1103/​PhysRevA.82.040303.

[208] Hannes Bernien, Sylvain Schwartz, Alexander Keesling, Harry Levine, Ahmed Omran, Hannes Pichler, Soonwon Choi, Alexander S Zibrov, Manuel Endres, Markus Greiner, et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature, 551, 2017. 10.1038/​nature24622.

[209] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac. Matrix product state representations. Quantum Info. Comput., 7 (5), Jul 2007. ISSN 1533-7146. 10.5555/​2011832.2011833.

[210] Olof Salberger and Vladimir Korepin. Fredkin spin chain. In Ludwig Faddeev Memorial Volume: A Life In Mathematical Physics. World Scientific, 2018. 10.1142/​9789813233867_0022.

[211] Ramis Movassagh. Entanglement and correlation functions of the quantum motzkin spin-chain. Journal of Mathematical Physics, 58 (3), 2017. 10.1063/​1.4977829.

[212] Libor Caha and Daniel Nagaj. The pair-flip model: a very entangled translationally invariant spin chain. arXiv, 2018. 10.48550/​arXiv.1805.07168.

[213] Khagendra Adhikari and K. S. D. Beach. Deforming the fredkin spin chain away from its frustration-free point. Phys. Rev. B, 99, Feb 2019. 10.1103/​PhysRevB.99.054436.

[214] Colin P. Williams. Explorations in Quantum Computing, Second Edition. Springer, 2011. 10.1007/​978-1-84628-887-6.

[215] Peter BR Nisbet-Jones, Jerome Dilley, Annemarie Holleczek, Oliver Barter, and Axel Kuhn. Photonic qubits, qutrits and ququads accurately prepared and delivered on demand. New Journal of Physics, 15 (5), 2013. 10.1088/​1367-2630/​15/​5/​053007.

[216] C. Senko, P. Richerme, J. Smith, A. Lee, I. Cohen, A. Retzker, and C. Monroe. Realization of a quantum integer-spin chain with controllable interactions. Phys. Rev. X, 5, Jun 2015. 10.1103/​PhysRevX.5.021026.

[217] Barry Bradlyn, Jennifer Cano, Zhijun Wang, MG Vergniory, C Felser, Robert Joseph Cava, and B Andrei Bernevig. Beyond dirac and weyl fermions: Unconventional quasiparticles in conventional crystals. Science, 353 (6299), 2016. 10.1126/​science.aaf5037.

[218] A Klümper, A Schadschneider, and J Zittartz. Matrix product ground states for one-dimensional spin-1 quantum antiferromagnets. EPL (Europhysics Letters), 24 (4), 1993. 10.1209/​0295-5075/​24/​4/​010.

[219] Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal Tasaki. Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett., Aug 1987. 10.1103/​PhysRevLett.59.799.

[220] Ian Affleck, Tom Kennedy, Elliott H Lieb, and Hal Tasaki. Valence bond ground states in isotropic quantum antiferromagnets. In Condensed matter physics and exactly soluble models. Springer, 1988. 10.1007/​978-3-662-06390-3_19.

[221] K. Wierschem and K. S. D. Beach. Detection of symmetry-protected topological order in aklt states by exact evaluation of the strange correlator. Phys. Rev. B, 93, Jun 2016. 10.1103/​PhysRevB.93.245141.

[222] Frank Pollmann, Erez Berg, Ari M. Turner, and Masaki Oshikawa. Symmetry protection of topological phases in one-dimensional quantum spin systems. Phys. Rev. B, 85, Feb 2012. 10.1103/​PhysRevB.85.075125.

[223] Sergey Bravyi, Libor Caha, Ramis Movassagh, Daniel Nagaj, and Peter W. Shor. Criticality without frustration for quantum spin-1 chains. Phys. Rev. Lett., 109, Nov 2012. 10.1103/​PhysRevLett.109.207202.

[224] Zhao Zhang, Amr Ahmadain, and Israel Klich. Novel quantum phase transition from bounded to extensive entanglement. Proceedings of the National Academy of Sciences, 114 (20), 2017. 10.1073/​pnas.1702029114.

[225] Eleonora Nagali, Linda Sansoni, Lorenzo Marrucci, Enrico Santamato, and Fabio Sciarrino. Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding. Phys. Rev. A, 81, May 2010. 10.1103/​PhysRevA.81.052317.

[226] Harald Niggemann, Andreas Klümper, and Johannes Zittartz. Quantum phase transition in spin-3/​2 systems on the hexagonal lattice—optimum ground state approach. Zeitschrift für Physik B Condensed Matter, 104 (1), 1997. 10.1007/​s002570050425.

[227] S Alipour, S Baghbanzadeh, and V Karimipour. Matrix product representations for spin-(1/​2) and spin-(3/​2) spontaneous quantum ferrimagnets. EPL (Europhysics Letters), 84 (6), 2009. 10.1209/​0295-5075/​84/​67006.

[228] Julia M. Link, Igor Boettcher, and Igor F. Herbut. $d$-wave superconductivity and bogoliubov-fermi surfaces in rarita-schwinger-weyl semimetals. Phys. Rev. B, 101, May 2020. 10.1103/​PhysRevB.101.184503.

[229] MA Ahrens, A Schadschneider, and J Zittartz. Exact ground states of spin-2 chains. EPL (Europhysics Letters), 59 (6), 2002. 10.1209/​epl/​i2002-00126-5.

[230] Maksym Serbyn, Dmitry A Abanin, and Zlatko Papić. Quantum many-body scars and weak breaking of ergodicity. Nature Physics, 17 (6), 2021. 10.1038/​s41567-021-01230-2.

[231] Sanjay Moudgalya, Nicolas Regnault, and B. Andrei Bernevig. Entanglement of exact excited states of affleck-kennedy-lieb-tasaki models: Exact results, many-body scars, and violation of the strong eigenstate thermalization hypothesis. Phys. Rev. B, 98, Dec 2018a. 10.1103/​PhysRevB.98.235156.

[232] Sanjay Moudgalya, Stephan Rachel, B. Andrei Bernevig, and Nicolas Regnault. Exact excited states of nonintegrable models. Phys. Rev. B, 98, Dec 2018b. 10.1103/​PhysRevB.98.235155.

[233] Soonwon Choi, Christopher J. Turner, Hannes Pichler, Wen Wei Ho, Alexios A. Michailidis, Zlatko Papić, Maksym Serbyn, Mikhail D. Lukin, and Dmitry A. Abanin. Emergent SU(2) dynamics and perfect quantum many-body scars. Phys. Rev. Lett., 122, Jun 2019. 10.1103/​PhysRevLett.122.220603.

[234] Naoyuki Shibata, Nobuyuki Yoshioka, and Hosho Katsura. Onsager's scars in disordered spin chains. Phys. Rev. Lett., 124, May 2020. 10.1103/​PhysRevLett.124.180604.

[235] Cheng-Ju Lin and Olexei I. Motrunich. Exact quantum many-body scar states in the rydberg-blockaded atom chain. Phys. Rev. Lett., 122, Apr 2019. 10.1103/​PhysRevLett.122.173401.

[236] F. Troiani. Entanglement swapping with energy-polarization-entangled photons from quantum dot cascade decay. Phys. Rev. B, 90, Dec 2014. 10.1103/​PhysRevB.90.245419.

[237] Michael Zopf, Robert Keil, Yan Chen, Jingzhong Yang, Disheng Chen, Fei Ding, and Oliver G. Schmidt. Entanglement swapping with semiconductor-generated photons violates Bell's inequality. Phys. Rev. Lett., 123, Oct 2019. 10.1103/​PhysRevLett.123.160502.

[238] Jian-Wei Pan and Anton Zeilinger. Greenberger-Horne-Zeilinger-state analyzer. Phys. Rev. A, 57, Mar 1998. 10.1103/​PhysRevA.57.2208.

[239] János A Bergou. Discrimination of quantum states. Journal of Modern Optics, 57 (3), 2010. 10.1080/​09500340903477756.

[240] N. Bent, H. Qassim, A. A. Tahir, D. Sych, G. Leuchs, L. L. Sánchez-Soto, E. Karimi, and R. W. Boyd. Experimental realization of quantum tomography of photonic qudits via symmetric informationally complete positive operator-valued measures. Phys. Rev. X, 5, Oct 2015. 10.1103/​PhysRevX.5.041006.

[241] Carlton M Caves, Christopher A Fuchs, and Rüdiger Schack. Unknown quantum states: the quantum de finetti representation. Journal of Mathematical Physics, 43 (9), 2002. 10.1063/​1.1494475.

[242] A. Hayashi, M. Horibe, and T. Hashimoto. Mean king's problem with mutually unbiased bases and orthogonal latin squares. Phys. Rev. A., May 2005. 10.1103/​PhysRevA.71.052331.

[243] Oliver Schulz, Ruprecht Steinhübl, Markus Weber, Berthold-Georg Englert, Christian Kurtsiefer, and Harald Weinfurter. Ascertaining the values of ${{\sigma}}_{x}$, ${{\sigma}}_{y}$, and ${{\sigma}}_{z}$ of a polarization qubit. Phys. Rev. Lett., 90, Apr 2003. 10.1103/​PhysRevLett.90.177901.

[244] Berthold-Georg Englert, Christian Kurtsiefer, and Harald Weinfurter. Universal unitary gate for single-photon 2-qubit states. Physical Review A, 63, Feb 2001. 10.1103/​PhysRevA.63.032303.

[245] Cheng-Qiu Hu, Jun Gao, Lu-Feng Qiao, Ruo-Jing Ren, Zhu Cao, Zeng-Quan Yan, Zhi-Qiang Jiao, Hao Tang, Zhi-Hao Ma, and Xian-Min Jin. Experimental test of tracking the king problem. Research, 2019, Dec 2019. 10.34133/​2019/​3474305.

[246] T. B. Pittman, B. C. Jacobs, and J. D. Franson. Demonstration of nondeterministic quantum logic operations using linear optical elements. Phys. Rev. Lett., 88, Jun 2002. 10.1103/​PhysRevLett.88.257902.

[247] Stuart M Marshall, Alastair RG Murray, and Leroy Cronin. A probabilistic framework for identifying biosignatures using pathway complexity. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 375 (2109), 2017. 10.1098/​rsta.2016.0342.

[248] Stuart M Marshall, Cole Mathis, Emma Carrick, Graham Keenan, Geoffrey JT Cooper, Heather Graham, Matthew Craven, Piotr S Gromski, Douglas G Moore, Sara Walker, et al. Identifying molecules as biosignatures with assembly theory and mass spectrometry. Nature Communications, 12 (1), 2021. 10.1038/​s41467-021-23258-x.

[249] Matthias J Bayerbach, Simone E D'Aurelio, Peter van Loock, and Stefanie Barz. Bell-state measurement exceeding 50% success probability with linear optics. Science Advances, 9 (32), 2023. 10.1126/​sciadv.adf4080.

[250] D Blume. Few-body physics with ultracold atomic and molecular systems in traps. Reports on Progress in Physics, 75, mar 2012. 10.1088/​0034-4885/​75/​4/​046401.

[251] Daniel E. Parker, Xiangyu Cao, Alexander Avdoshkin, Thomas Scaffidi, and Ehud Altman. A universal operator growth hypothesis. Phys. Rev. X, 9, Oct 2019. 10.1103/​PhysRevX.9.041017.

[252] Mario Krenn, Robert Pollice, Si Yue Guo, Matteo Aldeghi, Alba Cervera-Lierta, Pascal Friederich, Gabriel dos Passos Gomes, Florian Häse, Adrian Jinich, Akshat Kumar Nigam, et al. On scientific understanding with artificial intelligence. Nature Reviews Physics, 2022. 10.1038/​s42254-022-00518-3.

[253] Terry Rudolph. Terry vs an ai, round 1: Heralding single-rail (approximate?) 4-ghz state from squeezed sources. arXiv, 2023. 10.48550/​arXiv.2303.05514.

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[10] Jakob S. Kottmann and Francesco Scala, "A Quantum Algorithmic Approach to Multiconfigurational Valence Bond Theory: Insights from Interpretable Circuit Design", arXiv:2302.10660, (2023).

[11] Terry Rudolph, "Terry vs an AI, Round 1: Heralding single-rail (approximate?) 4-GHZ state from squeezed sources", arXiv:2303.05514, (2023).

[12] Zeqiao Zhou, Yuxuan Du, Xu-Fei Yin, Shanshan Zhao, Xinmei Tian, and Dacheng Tao, "Optical Quantum Sensing for Agnostic Environments via Deep Learning", arXiv:2311.07203, (2023).

[13] Tareq Jaouni, Xiaoqin Gao, Sören Arlt, Mario Krenn, and Ebrahim Karimi, "Experimental Solutions to the High-Dimensional Mean King's Problem", arXiv:2307.12938, (2023).

[14] Jonas Landgraf, Vittorio Peano, and Florian Marquardt, "Automated Discovery of Coupled Mode Setups", arXiv:2404.14887, (2024).

[15] Kobra Mahdavipour, Farzam Nosrati, Stefania Sciara, Roberto Morandotti, and Rosario Lo Franco, "Generation of genuine multipartite entangled states via indistinguishability of identical particles", arXiv:2403.17171, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-21 19:14:11) and SAO/NASA ADS (last updated successfully 2024-06-21 19:14:12). The list may be incomplete as not all publishers provide suitable and complete citation data.